Oops, I thought your claim was about the consonant sound /b/ vs /v/. I had the British /bi:tə/ in my mind, and forgot that Americans used /beɪtə/, which I agree is closer to the American pronunciation if your 'ay's are not diphthongised.

Funny enough, I went to double-check the IPA and realized the textbook classical Attic should be reconstructed as /ɛ/, so /bɛːta/ anyway. Which is still closer to the American version as both are open front vowels.

BUT, I just went down a rabbit hole and found this video on the history of the letter: https://www.youtube.com/watch?v=KS5POB2rLsw

It turns out that while /bɛːta/ is the old academic reconstruction, statistical analyses of spelling mistakes from then shows that Athenians had already closed that vowel to /e:/ or even all the way to the modern /i:/ sound as early as 500 BC. So the how they spoke daily was even messier.